Necking deformation in non-linear elasticity

Abstract

We study the asymptotic behaviour of the minimizers of the variational problem:

min {∫^a_−a[ε²(u′)²(x)+W(u(x))]dx:u∈L¹(−a,a),u≥0,∫^a_−au(x)dx=m,u(−a)=u(a)=c},

where m/2a∈(α,β) and W is a non-negative, continuous and non-convex function, with W(u) = 0 iff u∈{α,β}.

We prove that the presence of the necking is related to the value c; more precisely, we have a “neck” if c<α and a “buldge” if c>β.